Energy Functionals in Scattering Theory and Inversion of Low Frequency Moments

Abstract

The first part of these lectures involves short statements of scattering problems in acoustics, electromagnetics and elasticity. The relative energy conservation theorems are derived from the corresponding governing equations. Then the energy functionals of interest in scattering theory are defined from the basic energy terms entering the relative conservation laws. That includes the differential, the scattering, the absorption and the extinction cross sections for acoustic, electromagnetic and elastic scattering problems. In the second part, it is shown how a knowledge of some generalized low frequency moments, expressing the scattering amplitude, can be used to recover the exact shape of any star shape polynomial surface. The idea is to relate a finite number of generalized moments generated by the leading low-frequency approximation to a set of particular combined spherical moments that appear as coefficients of an algebraic linear system, whose solution provides the coefficients of the scattering surface in spherical harmonics. This is done with the help of an inner product defined over the surface of the unit sphere with respect to an unknown positive surface measure. The special case of a second degree surface, which corresponds to an ellipsoid, is discussed in detail.